Stability of delayed oscillators subjected to digital PD control*

Stabilization of time-delayed oscillators via digital PD controller is analyzed. The system under analysis is described by the delay-differential equationwhere tj = jΔt with Δt being a sampling period for the digital controller. The point-delay term x(t – τ) is assumed to be inherently present in the governing equation of the uncontrolled system, while the term x(tj–1) is present due to the digital controller. Since the term x(tj–1) can be represented as a term with a piecewise linearly varying time delay, the system is time-periodic at period Δt. The stability analysis for the system is performed using the semi-discretization method. Case studies are presented for the stabilization of the turning process via digital PD controller.